For the purpose of illustration and to explain the proposed solver methodology, let us consider a simple one dimensional domain (\(\Omega \) ) with two subdomains (\(\mathcal {N}=2\) ) as shown in Fig. (REF ), where \(\Omega _1\)  and \(\Omega _2\)  represent the subdomains that share an interface \(\partial \Omega _{1,2}\) . In Fig. (REF ), the layout of the knot spans for both an even degree (\(p=2\) ) and odd degree (\(p=3\) ) are shown. For generality, we also introduce here an overlap layer \(\Delta _1\)  and \(\Delta _2\)  on each subdomain that represents the set of shared knot spans with its adjacent subdomain (for internal boundaries), and an optional augmented layer \(\delta _1\)  and \(\delta _2\)  that has a connotation similar to that of an overlap region in traditional DD schemes [1]}. Note that in order to reconstruct the input data in \(\Omega _i, \forall i \in [1,2]\) , the knot spans must mandatorily include \(\Delta _i\)  regions. This \(\Delta _i\)  overlap region is required by definition to maintain partition of unity of a B-spline curve in order to evaluate Equation (REF ). For generality, \(\Delta _i\)  represents the repeated knots along clamped global domain boundaries, and the shared knots between two subdomains in the unclamped interior boundaries.
For arbitrary degree \(p\) , the number of knot spans in \(\Delta _i\)  is given by \(\left\lfloor \frac{p}{2} \right\rfloor \) , where \(\left\lfloor . \right\rfloor \)  represents the floor operator. In multidimensional tensor product expansions, these shared spans are replaced by shared layers of knot spans along the subdomain interfaces. The \(\delta _i\)  regions are additional, and optional, shared knot spans that can help improve error convergence in a manner similar to overlap regions in DD methods used for PDE solvers [2]}, [1]}.
